中国人民大学数学科学研究院
Modeling Biological Phenomena by Parabolic PDEs and their Analysis
主办单位:中国人民大学数学科学研究院
June 7, 2019
Room 2218, Renmin University of China
Schedule:

Time:   June 72019

Venue:  Room 2218, Renmin University of China, Beijing

Friday morning (June 7)

8:00-8:20

Registration

8:20-8:30

Opening ceremony

Session I

Chair: 

8:30-9:15

Shangbing Ai, University of Alabama in Huntsville

The entry-exit function and relaxation oscillations in slow-fast systems

9:15-10:00

Tian Xiang, Renmin University of China

Strong effect of chemo-repulsion mechanism on the prevention of blow-up

10:00-10:30

Picture and Tea Break

Session II

Chair:

10:30-11:15

Sohei Tasaki, RIKEN Center for Biosystems Dynamics Research

Morphologies of Bacillus subtilis communities responding to environmental variation

11:15-12:00

Xiulan Lai, Renmin University of China

Mathematical modeling about the scheduling of VEGF and PD-1 inhibitors in combination cancer therapy

Friday afternoon (June 7)

Session III

Chair:

13:30-14:15

Eiji Yanagida, Tokyo Institute of Technology

Traveling waves in the logarithmic diffusion equation with a reaction

14:15-15:00

Kanako Suzuki, Ibaraki University

Spatial patterns of some reaction-diffusion-ODE systems

15:00-15:30

Tea Break

Session IV

Chair:

15:30-16:15

Fang Li, Sun Yat-sen University

The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries

16:15-16:45

Round Table: What direction will be important, interesting and successful in mathematical sciences

16:45-17:00

closing

17:20-19:20

Banquet

Title and Abstract:

The entry-exit function and relaxation oscillations in slow-fast systems

Shangbing Ai

University of Alabama in Huntsville, USA

Abstract: The entry-exit function for the phenomenon of delay of stability loss (Pontryagin delay) arising in some slow-fast planar systems plays a key role in establishing the existence of limit cycles that exhibit relaxation oscillations. In this talk, we discuss an elementary approach to study the entry-exit function and its applications to a broad class of predator-prey models on the existence of single and multiple relaxation oscillations. An application to the existence of periodic traveling wave solutions to a diffusive predator-prey model is also to be presented.

Mathematical modeling about the scheduling of VEGF and PD-1 inhibitors in combination cancer therapy

Xiulan Lai

Renmin University of China, China


Abstract: One of the questions in the design of cancer clinical trials with combination of two drugs is in which order to administer the drugs. This is an important question, especially in the case where one agent may interfere with the effectiveness of the other agent. In this talk I will address this scheduling question by mathematical modeling approaches, in a specific case where one of the drugs is anti-VEGF, which is known to affect the perfusion of other drugs. As a second drug we take anti-PD-1. Both drugs are known to increase the activation of anticancer T cells. Our simulations show that in the case where anti-VEGF reduces the perfusion, a non-overlapping schedule is significantly more effective than a simultaneous injection of the two drugs, and it is somewhat more beneficial to inject anti-PD-1 first.

The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries

Fang Li

Sun Yat-sen University, China

Abstract: We introduce and study a class of free boundary models with “nonlocal diffusion”, which are natural extensions of the free boundary models in [1] and elsewhere, where “local diffusion” is used to describe the population dispersal, with the free boundary representing the spreading front of the species. We show that this nonlocal problem has a unique solution defined for all time, and then examine its long-time dynamical behavior when the growth function is of Fisher-KPP type. We prove that a spreading-vanishing dichotomy holds, though for the spreading-vanishing criteria significant differences arise from the well known local diffusion model in [1].


Reference:[1] Y. Du, Z. Lin, Spreading-Vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal. 42 (2010) 377-405.

Spatial patterns of some reaction-diffusion-ODE systems

Kanako Suzuki

Ibaraki University, Japan


Abstract: We discuss mathematical systems of pattern formation phenomena, which arise, for example, from modeling of interactions between cellular processes such as cell growth, differentiation or transformation and diffusion signaling factors.

Spatially continuous systems become a single reaction-diffusion equation coupled with an ordinary differential equation. We show that all continuous spatially heterogeneous stationary solutions can be unstable in most cases. Therefore, even if it is a model of pattern formation, there is no stable spatial pattern, which is very different from classical reaction-diffusion models. Moreover, the diffusion-driven blowup can occur.

From a biological point of view, we introduce a system in heterogeneous environment. Then, we see that solutions are bounded and there are stable spatial patterns in some sense.

Morphologies of Bacillus subtilis communities responding to environmental variation

Sohei Tasaki

RIKEN Center for Biosystems Dynamics Research, Japan


Abstract: Bacterial communities exhibit a variety of growth morphologies in constructing robust systems under different environmental conditions. We review the diverse morphologies of Bacillus subtilis communities and their mechanisms of self-organization. B. subtilis uses different cell types to suit environmental conditions and cell density. The subpopulation of each cell type exhibits various environment-sensitive properties. Furthermore, division of labor among the subpopulations results in flexible development for the community as a whole. We review how B. subtilis community morphologies and growth strategies respond to environmental perturbations.

Strong effect of chemo-repulsion mechanism on the prevention of blow-up

Tian Xiang

Renmin University of China, China


Abstract: We study essentially a Keller-Segel type chemotaxis system with nonlinear sensitivity (signifying by the exponent alpha) and production of signal (signifying by the exponent beta). We establish explicit relations between alpha, beta and the space dimension to ensure global- and local-in-time boundedness of classical solutions. In the attractive chemotaxis setting, our results cover the separate cases in the existing literature and they are critical by the quite known blow-up results. In the repulsive chemotaxis setting, we find that much wider regimes of alpha and beta compared to attraction case can ensure global existence and boundedness. Hence, our findings reveal strong “damping” effect of chemo-repulsion mechanism on boundedness, since blow-up has emerged if chemo-attraction mechanism is exerted instead.  Furthermore, our 3-D local-in-time boundedness moves one step further towards the popular saying that there would no blow-up in the 3-D minimal negative chemotaxis model. The achievement of our goal is based on a new qualitative boundedness criterion and a new uniform-in-time compound space-time bound for the gradient of the chemical concentration. Besides, our results also relax the regularity requirement on the initial chemical concentration in the existing literature. This is based on a joint with K. Lin from Southwest University of Finance and Economics.

Traveling waves in the logarithmic diffusion equation with a reaction

Eiji Yanagida

Tokyo Institute of Technology, Japan


Abstract: I will discuss the behavior of positive solutions to the logarithmic diffusion equation with a reaction term. First, the existence of traveling waves is studied by the phase plane analysis. It turns out that there are one-parameter family of traveling front solutions and two-parameter family of traveling pulse solutions. It is shown that the traveling front solution is asymptotically stable for a wide class of initial data, while the traveling pulse solution is unstable.